Amn Halloc Math Ecss E-mal : amn@sthths bpa : sthths/amn MATH EXERCISES GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR CONTINUITY AND NAVIER-STOKES EQUATIONS VECTOR RODUCTS I an thn scala o ot poct cto o s poct In som boos s also cons ot poct n b GRADIENT DIVERGENCE CURL DEL NABLA OERATOR LALACIAN OERATOR GRADIENT Lt ϕ b a scala l Th ant s th cto l n b a ϕ ϕ ϕ ϕ DIVERGENCE Lt R Q b a cto l contnosl ntabl th spct to an Thn th nc o s th scala l n b R Q CURL Th cl o s th cto l n b Q R Q R R Q cl o Q R Q R cl DEL NABLA OERATOR Th cto ntal opato s call l o nabla / 8
Amn Halloc Math Ecss U can not a an cl as blo: a ϕ ϕ cl Not that s not th sam as Q R LALACIAN OERATOR Th Laplacan opato s n o a scala l U b U U U U U an o a cto l Q R b Q R Som omlas o pola an clncal coonats ola coonats m ϑ ϑ ϑ ϑ tansomaton: aa lmnt: A stana bass: [Rma : Not that a pn on hn mo om pont to pont ths s th ason h ths bass n som boos s call local bass ] I n Catsan coo an ϑ th sam cto n pola coonats thn ϑ [Rma : V can ths omlas b calclatn th componnts o n th ctons o an Ths / 8
Amn Halloc Math Ecss smlal ϑ ϑ ] Clncal coonats : tansomaton: olm lmnt: V ϑ ϑ stana bass: I n Catsan coo an ϑ th sam cto n clncal coonats thn ha ollon cto componnts latonshp: ϑ [Rma : o ampl can t n th ollon a: ] scala l: ant: a laplacan: cto l: nc: cl: cl / 8
Amn Halloc Math Ecss EXERCISES n a b a an c cl o th ollon ntts a cl 0 o 0 b cl ϕ 0 o ϕ 0 Whch on o th ollon nctons a b ln c p 5 5 satss th Laplac qaton 0? n 5 Wt th nal tanspot qaton ϕ ϕu Γ ϕ S t φ thot opatos cl o a H U nctons ϕ Γ S a al nctons o t an 6 Whch on an o th ollon nctons a ϕ b ϕ ϕ c satss th qaton ϕu Γ ϕ S? H Γ 5 U an S 7 n hch on an o th ollon nctons a ϕ b ϕ 5 5 ϕ c satss th qaton / 8
Amn Halloc Math Ecss ϕ ϕu Γaϕ S t h Γ U an S 8 5 8 am 008 A Wt th nal tanspot qaton ϕ ϕu Γ ϕ S q t thot opatos cl o a H U nctons ϕ Γ S a al nctons o t an B Lt Γ U n S n th qaton q no that th ncton ϕ satss th qaton 9 Q6 am 008 Cons th ollon qaton ϕ ϕu Γ ϕ U 6 q t Lt Γ constant U n th constant Γ n th qaton q no that th ncton ϕ t satss th qaton 0 I possbl n o th n patal ats an a an b an c an an 5 Hnt: Ncssa conton: I has contns ats thn th m ats o shol b qal Ths * s th ncssa conton o th stnc o a ncton that has th n ats Dtmn th al o a o hch th sstm o patal ntal qatons 5 / 8
Amn Halloc Math Ecss a an has soltons Thn n ponn to ths al o a I possbl n o th n patal ats an a an b an c an an Hnt: Ncssa conton: I has contnos ats thn th m ats o shol b qal Ths Con : Con : Con : a th ncssa conton o th stnc o a ncton that has th n ats Dtmn th als o a an b o hch th sstm o patal ntal qatons a an b has soltons Thn n ponn to ths als o a an b W cons an ncompssbl nst const sta stat aabls o not pn on tm sothmal Ntonan lo th a n loct l V Us th ollon qatons contnt an Na Stos qatons to n n psson o pss as a ncton o an h constant µ constant 00 0 an h 98m / s 6 / 8
Amn Halloc Math Ecss Incompssbl contnt qaton: 0 q Na Stos qatons: componnt: µ q t componnt: µ q t componnt: µ q t a V 0 b V c V 5 am 009 A Cons th ollon qaton ϕ ϕu Γ ϕ U 6 6 8 t Lt Γ constant U n th constant Γ n th qaton q no that th ncton ϕ t satss th qaton q B W cons an ncompssbl nst const sta stat aabls o not pn on tm sothmal Ntonan lo th a n loct l V Us th ollon qatons contnt an Na Stos qatons to n n psson o pss as a ncton o an h constant µ constant 00 0 an h 98m / s an V 6 6 am 009 W cons an ncompssbl nst const sta stat aabls o not pn on tm sothmal Ntonan lo th a n loct l V Us th ollon qatons contnt an Na Stos qatons to n st paamt a an thn 7 / 8
Amn Halloc Math Ecss n psson o pss as a ncton o an h constant µ constant 00 0 an h 98m / s an V 5 a 7 Cons sta ncompssbl sothmal lamna statona Ntonan lo n a lon on pp n th -cton th constant ccla s-scton o as R m Us th contnt an th Na-Stos qatons n clncal coonats to n th loct l V an th pss l th l lo satss th ollon contons: c0 All patal ats th spct to tm t a 0 Sta lo c μ000 /m s an 000 /m c A Constant pss ant / /50 a/m s appl n th hoontal as -as n o notaton: / /50 c Th lo s paalll to th as that s 0 an 0 c W assm that th lo s asmmtc Th loct os not pn on that s 0 c5 Bona con No-slp bona conton V l V all : I thn 0 c6 Bona conton : has mamm at 0 that s 0 0 --------------------------------------------------------------------------------------------- Th contnt an th Na-Stos qatons o an ncompssbl sothmal Ntonan lo nst const st µ const th a loct l V n Clncal coonats : Incompssbl contnt qaton 0 q a Na-Stos qatons n Clncal coonats: -componnt: t µ -componnt: q b 8 / 8
Amn Halloc Math Ecss t µ q c -componnt: t µ q 8 Eam Mach 0 qston A ponts W cons an ncompssbl nst const sta stat aabls o not pn on tm sothmal Ntonan lo th a n loct l a b c V Us th ollon qatons contnt an Na Stos qatons h constant µ constant 00 0 an / 98 s m to n: paamts a b an c n psson o pss as a ncton o an Th GRADIENT VECTOR th chan o aabls an bass Th ant cto o th ncton s n as a * I chan aabls to an plac bass ctos th n lnal npnnt ctos thn can pss th sam ant cto a n tms o aabls an ctos W smpl calclat th ats an n n aabls an pss as a lna combnatons o Thn sbsttt thos als nto * S th ollon ampl 9 W cons a scala l n n clncal coonats h an bass ctos a 9 / 8
Amn Halloc Math Ecss n th psson o th ant a n clncal coonats that s n tms o an a p th sam bass b c am 0; Q5 B ponts ths s otn s as a local bass o clncal coonats 0 am 06; Q6 A ponts D th Cach momntm qaton DV σ Dt n th l o th cto l pa thoh th sac 0 0 Us th Dnc Thom to n th l o th cto l ot o th sph S th qaton 9 ANSWERS AND SOLUTIONS: Solton: Q R a Snc ha 0 0 Ans a ϕ ϕ ϕ b Snc a ϕ ha o ϕ a 00 Ans b a 00 c cl Q R 0 / 8
Amn Halloc Math Ecss Ans c cl Hnt Us th ntons o an cl 0 0 Ans: Th nctons ln an 5 5 satss th Laplac qaton Ans: cl a 6 5 Solton: ϕ ϕu Γaϕ Sφ t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ Sφ t ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ t 6 Whch on an o th ollon nctons a ϕ b ϕ ϕ c satss th qaton ϕu Γ ϕ S? H Γ 5 U an S Solton : Th qaton ϕu Γ ϕ S can b ttn as ϕ Γ S φ / 8
Amn Halloc ϕu Γaϕ S 5ϕ 5ϕ 5ϕ ϕ ϕ ϕ ϕ ϕ ϕ 5 ϕ 5 ϕ 5 ϕ Math Ecss q a Lt ϕ ϕ V calclat th ats oϕ an sbsttt n th lt han s LHS an ht han s o th qaton q ϕ ϕ ϕ ϕ 5 ϕ 5 ϕ LHS: 5 RHS 60 Whnc LHS RHS Ths th ncton ϕ s not a solton to th qaton b ϕ ϕ LHS RHS Whnc LHS RHS an th ncton ϕ s not a solton to th qaton ϕ ϕ c Lt Thn LHS 6 RHS 7 Ths LHS RHS an th ncton ϕ s not a solton to th qaton Ans: Non o th nctons satss th qaton 7 Ans: ncton ϕ 5 satss th qaton 8 am 98 A Wt th nal tanspot qaton ϕ ϕu Γ ϕ S q t thot opatos cl o a H U nctons ϕ Γ S a al nctons o t an / 8
Amn Halloc Math Ecss B Lt Γ U n S n th qaton q no that th ncton ϕ satss th qaton Solton: A ϕ ϕu Γ ϕ S t ϕ ϕu Γaϕ S t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ S t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ S q t B W sbsttt Γ U an ϕ n th qaton q an t ϕ ϕ 8φ ϕ ϕ ϕ 0 S 0 8 0 6 8 S Consqntl S 8 8 9 Q6 am 008 Cons th ollon qaton ϕ ϕu Γ ϕ U 6 q t Lt Γ constant U n th constant Γ n th qaton q no that th ncton ϕ t satss th qaton Solton: ϕ ϕu Γ ϕ U 6 t ϕ ϕu Γaϕ cl U 6 t c cl U 0 ha cl U 0 / 8
Amn Halloc Math Ecss ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ 0 6 t ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ 0 6 q t W sbsttt U an ϕ t n th qaton q an t ϕ ϕ ϕ φ ϕ ϕ ϕ Γ Γ Γ 6 t Not that Γ s a constant 6 0 Γ Γ 6 8 Γ Γ Ans: Γ 0 I possbl n o th n patal ats an a an b an c an an 5 Hnt: Ncssa conton: I has contnos ats thn th m ats o shol b qal * s th ncssa conton o th stnc o a ncton that has th n ats Ans: a C b C c C No solton c th conton * s not lll 5 Solton a Snc an th ats a contnos th conton * s lll an can n o th n ats In o to n ntat th spct to th st o th qatons q / 8
Amn Halloc Math Ecss an t q C Ths C W ha ntat th spct to tho th constant stll pn on No to n C ntat an sbsttt n q an t: C C C C C nall sbstttn C C n ha C h C s a constant Ans: om a a Thn o a ha C Ans: a C b C c C No solton c th conton Con s not lll Solton a a an Snc th contons Con a lll an can n o th n ats In o to n ntat th spct to th st o th qatons q q q an t C Ths 5 / 8
Amn Halloc Math Ecss C W ha ntat th spct to tho th constant stll pn on an No to n C ntat an sbsttt n q an t: C C C C C W ha ntat th spct to tho th constant stll pn on an Ths C No sbstttn n q ha C C C C C nall sbstttn C C n ha C h C s a constant Ans: om a a b b b b Ths all th contons a lll a an b o ths als o a an b t C Calclaton o th pss l o a non loct l o an ncompssbl sta stat sothmal Ntonan lo Ans: a 8 8 C 6 / 8
Amn Halloc Math Ecss 7 7 b C c C Solton a W sbsttt 0 n q an t not that al ats th spct to t a 0: Contnt qaton: 0 0 q ntcall lll Na Stos qatons: componnt: 6 q componnt: 6 q componnt: 0 q No q s 8 C * Sbsttton n q mpls C 6 C 8 C Hnc om * ha 8 8 C ** No sbsttt ** n q an t 0 0 C C C h C s a constant nall sbstttn C C n ** ha 8 8 C h C s a constant 5 Solton A: ϕ ϕ U Γ ϕ U 6 6 8 t ϕ ϕu Γaϕ cl U 6 6 8 t c cl U 0 ha cl U 0 ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ 6 6 8 t 7 / 8
Amn Halloc Math Ecss ϕ ϕ ϕ ϕ ϕ ϕ ϕ Γ Γ Γ 6 6 8 t q W sbsttt U an ϕ t n th qaton q an t ϕ 8ϕ 8ϕ φ ϕ ϕ Γ Γ t Not that Γ s a constant 6 6 6 8 6Γ 6 6 8 0 6Γ Γ 5 Ans A: Γ 5 Solton B: W sbsttt 6 n q an t not that al ats th spct to t a 0: Contnt qaton: 0 0 q ntcall lll Na Stos qatons: componnt: 6 q componnt: 8 q componnt: q No q s 8 C * Sbsttton n q mpls C 8 C 8 C Hnc om * ha 8 8 C ** No sbsttt ** n q an t C C C h C s a constant nall sbstttn C C n ** ha 8 8 C Ans B: Γ ϕ 6 6 8 8 / 8
Amn Halloc Math Ecss C 8 8 h C s a constant 6 Solton 5 a V st sbsttt a 5 n q an t not that al ats th spct to t a 0: Contnt qaton: 0 a a No ha 5 V U th Na Stos qatons t: componnt: 6 9 q componnt: 5 q componnt: q No q s 6 9 C * Sbsttton n q mpls 5 5 C C C Hnc om * ha 5 6 9 C ** W sbsttt ** n q an t C C C h C s a constant nall sbstttn C n ** ha C 5 6 9 Ans : C 5 6 9 h C s a constant Q7 Cons sta ncompssbl sothmal lamna statona Ntonan lo n a lon on pp n th -cton th constant ccla s-scton o as R m Us 9 / 8
Amn Halloc Math Ecss th contnt an th Na-Stos qatons n clncal coonats to n th loct l V an th pss l th l lo satss th ollon contons: c0 All patal ats th spct to tm t a 0 Sta lo c μ000 /m s an 000 /m c A Constant pss ant / /50 a/m s appl n th hoontal as -as n o notaton: / /50 c Th lo s paalll to th as that s 0 an 0 c W assm that th lo s asmmtc Th loct os not pn on that s 0 c5 Bona con No-slp bona conton V l V all : I thn 0 c6 Bona conton : has mamm at 0 that s 0 0 Th contnt an th Na-Stos qatons o an ncompssbl sothmal Ntonan lo nst const st µ const th a loct l V n Clncal coonats : ---------------------------------------------------------------- SOLUTION Incompssbl contnt qaton 0 q a Na-Stos qatons n Clncal coonats: -componnt: t µ q b -componnt: t µ q c -componnt: 0 / 8
Amn Halloc Math Ecss t µ q W choos as a tcal as an a n a hoontal plan an th lo s paalll th th -as W not loct cto V h an a -componnt - componnt an -componnt n clncal coonats Accon to th assmptons ha 0 0 an os not pn on Snc s th tcal as ha that cto - 00 h 98 m/s hch n clncal coonats s an 0 No sbsttt / /50 a/m μ000 /ms n th contnt an Na- Stos qatons: Snc 0 an 0 accon to c contnt qaton n clncal coonats 0 s 0 / 8
Amn Halloc Math Ecss Ths tlls s that s not a ncton o thmo c loct os not pn on assmpton c concl that pns onl on To smpl notaton not * No sbsttt an 0 / /50 a/m μ000 /ms n th Na-Stos qatons: Th -componnt o th Na-Stos qaton s: 0 q -c Th -componnt o th Na-Stos qaton: 0 q -c Th Z-componnt o th Na-Stos qaton h an 0 q -c 50 000 50 s: Stp W n th pss In o to n th pss sol q -c q -c an th qaton s 50 that 50 om ths qatons t C 50 / 8
Amn Halloc Math Ecss Stp W n th loct componnt W sol q -c th bonas c5 an c6: 0 q -c 50 000 0 c5 0 0 c6 Rma: Tchncall can t nsta aabl om q -c ha 0 50 000 c s no a ncton o onl on C sbsttton 0 an c6 C 0 C sbsttton an c5 C Ths an V 0 0 Ans : C 50 V 0 0 / 8
Amn Halloc Math Ecss 8 W cons an ncompssbl nst const sta stat aabls o not pn on tm sothmal Ntonan lo th a n loct l a b c V Us th ollon qatons contnt an Na Stos qatons h constant µ constant 00 0 an / 98 s m to n: paamts a b an c n psson o pss as a ncton o an Incompssbl contnt qaton: 0 q Na Stos qatons: componnt: t µ q componnt: t µ q componnt: t µ q ------------------------------------------------------------- W sbsttt a b c n q an t not that al ats th spct to t a 0: Contnt qaton: 0 a a q Ths b c Na Stos qatons: componnt: q componnt: q componnt: / 8
Amn Halloc Math Ecss q Th sstm q q q s solabl onl m ats a qal: : c c Con 0 0 : b bc Con 0 0 : b b Con Ths c an b0 W sol smpl qatons an t C 6 8 5 Ans C 6 8 5 9 W cons a scala l n n clncal coonats h an bass ctos a n th psson o th ant a n clncal coonats that s n tms o an a p th sam bass b c am 0; Q5 B ponts ths s otn s as a local bass o clncal coonats 5 / 8
Amn Halloc Math Ecss Solton: In aabls ha a q o clncal coonats ha st t th ats an n coonats th aabl s n both coo sstms Soln th ollon sstm o an t ** W sbsttt th ats ** n q an t *** a To sol poblms a b c an mst pss as a lna combnatons o an sbsttt thm nto *** a om *** c ha mmatl a b om n q b Thn pt om q b nto *** an t a 6 / 8
Amn Halloc Math Ecss an at collctn componnts o a c om ha q c ttn om q c nto *** s a W can sol n th sam mann as n abc bt ths tm can st collct tms an t th slt: a Ans: a a b a c a a 0 S http://nomhannths/amn/ar_000/hl008/deriv_navier_stokesp Solton: N 7 / 8
Amn Halloc Math Ecss N Φ N Φ D Ans: Φ 0 0 0 Solton: Φ ΦV K Ans: K Φ π VolmK π π 8 / 8